Triangle Partitioning Lightoj 1043

 We know that,

Area of triangle ABC=sqrt(s*(s-ab)*(s-ac)*(s-bc));

so, we can find the height of the triangle by area=1/2 * base*height

Then we will find the position of DE by doing binary search. Then we will find the Distance A to D. 

 /// Bismillahir Rahmanir Rahim
/* Mohammad Morsalin
   Dept of ICE, NSTU
*/
#include<bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;
#define ll long long
#define pb push_back
#define mp make_pair
#define endl "\n"
#define int long long
#define f0(n) for(int i=0;i<n;i++)
#define ms(x) memset(x,0,sizeof(x))
#define ms2d(x,m,n) memset(x, 0, sizeof(x[0][0]) * m * n)
#define uniq(vec) vec.resize(distance(vec.begin(),unique(vec.begin(),vec.end())))
#define ordered_set tree<int, null_type,less<int>, rb_tree_tag,tree_order_statistics_node_update>
#define pi pair<int,int>
#define tc(t) int t;cin>>t;while(t--)
#define bits(n) __builtin_popcount(n)
#define maxpq priority_queue<int>
#define minpq priority_queue<int, vector<int>, greater<int> >
#define ins insert
#define ALL(v) v.begin(),v.end()
#define highest(x) numeric_limits<x>::max()
#define lowest(x) numeric_limits<x>::min()
#define Inf INFINITY
#define minv(v) *min_element(v.begin(),v.end())
#define maxv(v) *max_element(v.begin(),v.end())
#define fi first
#define se second
#define PI acos(-1)
#define sz(a) (int)a.size();
#define IOS ios::sync_with_stdio(false);
using namespace std;
int gcd(int a, int b) { if (a == 0) return b; return gcd(b % a, a);}
typedef tree<pair<int, int>,null_type,less<pair<int, int>>,rb_tree_tag,tree_order_statistics_node_update> ordered_multiset;
int dx8[] = {0, 0, 1, 1, 1, -1, -1, -1};
int dy8[] = {1,-1, 1, -1, 0, 0, -1, 1};
int dx4[] = {0, 0, 1, -1};
int dy4[] = {1, -1, 0, 0};
const long long MOD = 1000000007;
double sq(double x) {return x*x;}
template<typename T>inline T Bigmod(T base, T power, T MOD){
    T ret=1;
    while(power)
    {
        if(power & 1)ret=(ret*base)%MOD;
        base=(base*base)%MOD;
        power>>=1;
    }
    return ret;
}

bool sortinrev(const pair<int,int> &a,
               const pair<int,int> &b)
{
       return (a.first > b.first);
}
double const range=1e-10;

signed main()
{
    IOS
    int tn=0;
    tc(t){

        double ab,ac,bc,rat;
        cin>>ab>>ac>>bc>>rat;
        double s=(ab+bc+ac)/2;
        double ABC=sqrt(s*(s-ab)*(s-ac)*(s-bc));
        double theta=asin(ABC/(ab*bc));
        double highet=(ABC*2)/bc;
        double DEBC=ABC/(rat+1);
        double ADE=(ABC-DEBC);
        double low=0,high=bc,H=0,de;
        while(fabs(highet-H)>range){
            de=(low+high)/2;
            H=((2*ADE)/de)+((2*DEBC)/(de+bc));
            if(H>highet+range) low=de;
            else high = de;
        }
        double ad=(ADE/(de*sin(theta)));
        cout << "Case "<< ++tn << ": ";
        cout << fixed<<setprecision(10) << ad << endl;
       // printf("Case %d: %0.10lf\n",++tn,ad);
    }
    return 0;
}
///Alhamdulillah